For a random walk in an elliptic i.i.d. random environment in dimension
greater than or equal to 4, satisfying the ballisticity condition (T'), We
consider the probability of linear slowdown. We show an upper bound for this
probability which is very close to the lower bound obtained by the "naive
trap" analysis.
As a tool for obtaining the main result, we show an almost local version of
the quenched central limit theorem under the assumption of condition (T').